Unstable resonators are traditionally applied to high-power lasers, and the self-collimated unstable resonator laser to be described herein is intended for use in applications which require high power in the range from about 0.5 W to at least about 10 W.
There is a pressing need in various applications for high-power, single-lateral-mode diode lasers; they include pumps for optical fiber amplifiers in telecommunications, pumps for solid-state lasers and direct sources for space optical communications. Heretofore, only phased arrays have approached the necessary power for these applications while maintaining single-lateral mode. A diode laser that could demonstrate even half a watt continuous power in a narrow collimated beam would be in great demand in the field of telecommunications alone.
In the past, all but one of the known unstable resonant semiconductor lasers have used diverging and/or flat mirrors for the resonant cavity. This results in a strongly diverging beam that must be externally collimated with a large numerical aperture lens to achieve a narrow diffraction limited beam. The only known unstable resonant laser with a collimated beam utilizes a confocal structure in which the light is coupled out through a separate facet, as shown in FIGS. 1a and 1b. See J. Salzman, R. Lang, A. Larson and A. Yariv, "Confocal unstable-resonator semiconductor laser," Optics Letters, Vol. 11, No. 8, pp. 507-509 (August 1986). The result is two beams, each of half power separated by the totally reflecting convex mirror M.sub.2.
In semiconductor lasers, a Fabry-Perot cavity is normally provided using two flat parallel mirrors (cleaved surfaces), a condition which lies on the boundary of g parameters between stable and unstable optical resonance, as shown in FIG. 2a from A. E. Siegman, "Unstable Optical Resonators for Laser Applications," Proc. IEE, Vol. 53, pp. 277-287 (March 1965). If the distance between the two mirrors is denoted by L, as shown in FIG. 2b, and the mirror radii of curvature of mirrors M.sub.1 and M.sub.2 by R.sub.1 and R.sub.2, any given resonator may be represented by a single point in the g.sub.1, g.sub.2 plane, such as that labeled PLANAR in FIG. 2a.
Using the convex mirror geometry illustrated in FIG. 3, g-parameter analysis shows that for the radii r.sub.1 and r.sub.2 of the beams reflected by cavity mirrors M.sub.1 and M.sub.2, the geometric magnification M can be determined as a function of the mirror radii R.sub.1 and R.sub.2 from a set of g parameters defined as ##EQU3## where R.sub.1 and R.sub.2 are normalized to the distance separating the centers of the mirrors. Then, according to Siegman's formalism, the centers of curvature of the beams and geometric magnification are given by: ##EQU4## This g-parameter analysis applies to the concave mirror geometry of FIG. 2b and the concave-convex mirror geometry of FIG. 1b as well.
The most important parameter in the design of an unstable resonator is the magnification M, which, in the limit of a large Fresnel number, determines the cavity losses and thus the required threshold gain. When designing an unstable resonator, it is desirable to begin with the value of M, the specification of the magnification, and derive the necessary mirror radii R.sub.1 and R.sub.2 with the additional degree of freedom being determined by some other requirement, e.g., the mirror or beam radii. Positive radius R.sub.1 or R.sub.2 corresponds to curvature of the mirror M.sub.1 or M.sub.2 toward the resonator cavity, i.e., to a concave mirror, while a negative radius value corresponds to a convex mirror. Inverting Equations (1)-(3) to solve for the mirror radii as a function of M, however, is difficult. To simplify the mathematics, an alternate set of parameters .alpha. and .beta. are introduced, for which expressions for mirror and beam radii are easily derivable from the magnification and which yield simple, easily manipulable expressions. The alternate set of parameters are defined as: ##EQU5## Then algebraic substitution of Equation (4) into Equations (1)-(3) yields the following expressions: ##EQU6## According to the definition in Equation (4), .alpha. must be positive and real to achieve an unstable resonator (imaginary .alpha. corresponds to a stable resonator), while .beta. can take on any real value.
To put this to practical use, choose a magnification M, calculate .alpha. from Equation (6) and then by varying the parameter .beta., see the effects on mirror and beam radii. As an example, calculate the condition for self-collimation of an unstable resonator made of a material with index of refraction denoted by n. FIG. 4 is a ray diagram showing the refraction of a ray as it goes through a mirror of radius R.sub.2 and is subsequently collimated. From the diagram it is seen that in the paraxial approximation (where sin.theta..apprxeq..theta.), a ray traveling at angle .phi. with the optical axis hits the mirror M.sub.2 at an angle (.theta.-.phi.) from the normal; as the ray exits the medium having an index of refraction n, this will be magnified by Snell's law to an angle n(.theta.-.phi.) which should be equal to .theta. in order to be collimated. This relation is then combined with the relations between .theta., .phi., and the geometry of the resonator ##EQU7## to get the condition for self-collimation: ##EQU8## A geometric construction similar to FIG. 3 would show that Equation (9) holds for a concave mirror having negative values of R.sub.2 as well.
Expressing the relationship of Equation (9) in terms of g-parameters gives the following complex algebraic expression that is difficult to solve: ##EQU9## However, expressing this relationship in terms of .alpha. and .beta. gives a much simpler condition: EQU .beta.=.alpha.n. (11)
The index of refraction n is determined by the material through which the light passes out of the laser, so for an GaAs/AlGaAs laser, for example, n is the effective index of the waveguide heterostructure, about 3.4. Substituting Equation (11) into Equations (6) and (7) gives mirror and beam radii for the self-collimating geometry. They will be functions solely of .alpha., which is directly related to the magnification M and index of refraction n. ##EQU10## If r.sub.1 or r.sub.2 lie in the interval [-1,0], then the resonator has a focal spot between the mirrors. This is undesirable because high optical powers can cause self-focusing and/or optical damage at the focal spot. Examination of Equation (13) reveals that EQU r.sub.1 .epsilon.[-1,0]for.alpha..epsilon.[1/n,1]r.sub.2 .epsilon.[-1,0]for.alpha.&gt;1/n. (14)
Comparing these conditions on .alpha. to the relation between .alpha. and the magnification M from Equation (5), shows that there will always be at least one internal focal spot unless .alpha.&lt;1/n, or ##EQU11##
For a typical AlGaAs waveguide (n=3.4), the critical value of magnification occurs at M=1.833, a value determined by the index of refraction of the laser medium above which there is no self-collimating geometry possible without an internal focal spot. However, for lower magnifications, a self-collimating unstable resonator is indeed possible, as will be described below as an example of the present invention with reference to FIGS. 5a and 5b.
A new formalism for geometric analysis of unstable resonators has been presented which yields the self-collimation condition of an unstable semiconductor resonator. Such a laser would, in principle, have a collimated, diffraction-limited output beam and be capable of high output power. It should be noted at the outset that while a specific example of the invention is described below with reference to a planar (2-dimensional) unstable resonator laser, the principles are easily extendable to 3-dimensional unstable resonator lasers.